CHAPTER 4 Discount Rates as a Function of Log Size 155
APPENDIX A
Automating Iteration using NewtonÔÇ™s Method
This appendix is optional. It is mathematically dif´¬ücult and is more an-
alytically interesting than practical.
In this section we present a numerical method for automatically it-
erating to the correct log size discount rate. Isaac Newton invented an
iterative procedure using calculus to provide numerical solutions to equa-
tions with no analytic solution. Most calculus texts will have a section on
his method (for example, see Thomas 1972). His procedure involves mak-
ing an initial guess of the solution, then subtracting the equation itself
divided by its own ´¬ürst derivative to provide a second guess. We repeat
the process until we converge to a single answer.
The bene´¬üt of NewtonÔÇ™s method is that it will enable us to simply
enter assumptions for the cash ´¬‚ow base and the perpetual growth, and
the spreadsheet will automatically calculate the value of the ´¬ürm without
our having to manually go through the iterations as we did in Tables
4-4A, B, and C. Remember, some iteration process is necessary when
using log size discount rates because the discount rate is not independent
of size, as it is using other discount rate models.
To use NewtonÔÇ™s procedure, we rewrite equation (4-13) as:
CF
Let f(V) V 0 (A4-1)
(a b ln V g)
bCF
f (V) 1 (A4-2)
g)2
V(a b ln V
Assuming our initial guess of value is V0, the formula that de´¬ünes our
next iteration of value, V1, is:
CF
V0
(a b ln V0 g)
V1 V0 (A4-3)
bCF
1
g)2
V0(a b ln V0
Table A4-5 shows NewtonÔÇ™s iterative process for the simplest valu-
ation. In B22ÔÇ“B26 we enter our initial guess of value of an arbitrary $2
trillion (B22), our forecast cash ´¬‚ow base of $100,000 (B23), perpetual
growth g 7% (B24), and our regression coef´¬ücients a and b (B25 and
B26, which come from Table 4-1, E42 and E48, respectively).
In B7 we see our initial guess of $2 trillion. The iteration #2 value
of $280,530 (B8) is the result of the formula in the note immediately below
Table A4-5, which is equation (A4-3).33 B9 to B12 are simply the formula
in B8 copied to the remaining spreadsheet cells.
Once we have the formula, we can value any ´¬ürm with constant
growth in its cash ´¬‚ows by simply changing the parameters in B23 to
B24.

B31 to B34 show the sensitivity of the model to the initial guess. If
we guess poorly enough, the model will explode instead of converging
to the right answer. For this particular set of assumptions, an initial guess
of anywhere between $27,000 and $2 trillion will converge to the right
answer. Assumptions above $3 trillion or below $26,000 explode the
model.
Unfortunately, the midyear Gordon model, which is more accurate,
has a much more complex formula. The iterative process does converge,
but much too slowly to be of any practical use. One can use the end-of-
year Gordon model and multiply the result by the square root of (1 r).

CHAPTER 4 Discount Rates as a Function of Log Size 157
APPENDIX B
Mathematical Appendix
This appendix provides the mathematics behind the log size model, as
well as some philosophical analysis of the mathematicsÔÇ”speci´¬ücally on
the nature of exponential decay function and how that relates to phenom-
ena in physics as well as our log size model. This is intended more as
intellectual observation than as required information.
We will begin with two de´¬ünitions:
r return of a portfolio
S standard deviation of returns of the portfolio
Equation (B4-1) states that the return on a portfolio of securities (each
decile is a portfolio) varies positively with the risk of the portfolio, or:

r a1 b1S (B4-1)

This is a generalization of equation (4-1) in the chapter. This rela-
tionship is not directly observable for privately held ´¬ürms. Therefore, we
use the next equation, which is a generalization of equation (4-2) from
the chapter, to calculate expected return.
The parameter a1 is the regression estimate of the risk-free rate,34
while the parameter b1 is the regression estimate of the slope, which is
the return for each unit increase of risk undertaken, i.e., the standard
deviation of returns. Thus, b1 is the regression estimate of the price of or
the reward for taking on risk.

r a2 b 2 ln FMV, b 2 0 (B4-2)

Equation (B4-2) states that return decreases in a linear fashion with
the natural logarithm of ´¬ürm value. The parameter a2 is the regression
estimate of the return for a $1 ´¬ürm35 ÔÇ”the valueless ´¬ürmÔÇ”while the pa-
rameter b 2 is the regression estimate of the slope, which is the return for
each increase in ln FMV. Thus, b 2 is the regression estimate of the reduc-
tion in return investors accept for investing in smaller ´¬ürms. The terms
a1, a2, b1, and b 2 are all parameters determined in regression equations
(4-1) and (4-2).
Using all 73 years of stock market data, our regression estimate of
a1 6.56% (Table 4-1, D26), which compares well with the 73-year mean
Treasury Bond yield of 5.28%. It would initially appear that the log size
regression does a reasonable job of also providing an estimate of the risk-
free return. Unfortunately, it is not all that simple, as the log size estimate
using 60 years of data fares worse. The log size 60-year estimate of a1 is
8.90% (Table 4-1, E26), which is a long way off from the 60-year mean
treasury bond yield of 5.70% (Table 4-1, E27). Thus, eliminating the ´¬ürst

PART 2 Calculating Discount Rates
158
13 years of data had the effect of shifting the regression line upwards and
´¬‚attening it slightly.
We already knew from our analysis of Table 4-2 in the chapter that
using 60 years of data was the overall best choice because of its superi-
ority in the log size equation estimates, but it was not the best choice for
estimating equation (4-1). Its R 2 is lower and standard error is higher
than the 73-year results.
Focusing now on equation (B5-2), the log size equation, the 60-year
regression estimate of b1 1.0309% (Table 4-1, E48), which is signi´¬ü-
cantly lower in absolute value than the 73-year estimate of 1.284%
(D48). The parameter b 2 is the reduction in return that comes about from
each unit increase in company value (in natural logarithms). The param-
eter a2 is the y-intercept. It is the return (discount rate) for a valueless
´¬ürmÔÇ”more speci´¬ücally, a $1 ´¬ürm in valueÔÇ”as ln($1) 0.
Equating the right-hand sides of equation (B4-1) and (B4-2) and solv-
ing for S, we see how we are implicitly using the size of the ´¬ürm as a
proxy for risk.
a2 a1 b2
S ln FMV (B4-3)
b1 b1
Since a2 is the rate of return for the valueless ´¬ürm and a1 is the re-
gression estimate of the risk-free rateÔÇ”´¬‚awed as it isÔÇ”the difference be-
tween them, a2 a1 is the equity premium for a $1.00 ´¬ürm, i.e., the val-
ueless ´¬ürm. Dividing by b1, the price of risk (or reward) for each
increment of standard deviation, we get (a2 a1)/b1, the standard devi-
ation of a $1 ´¬ürm. We then reduce our estimate of the standard deviation
by the ratio of the relative prices of risk in size divided by the price of
risk in standard deviation, and multiply that ratio by the log of the size
of the ´¬ürm. In other words, we start with the maximum risk, a $1 ´¬ürm,
and reduce the standard deviation by the appropriate price times the log
of the value of the ´¬ürm in order to calculate the standard deviation of the
´¬ürm.
Rearranging equation (B4-3), we get
(a1 a2) b1S
ln FMV (B4-4)
b2
Raising both sides of the equation as powers of e, the natural exponent,
we get:
(a1 a2) b1S (a1 a2) b1S

FMV e e e b , or (B4-5)
b2 b2 2

b1
(a1 a2)
kS
FMV Ae , where A e ,k 0 (B4-6)
b2
b2
Here we see that the value of the ´¬ürm or portfolio declines exponentially
with risk, i.e., the standard deviation.
Unfortunately, the standard deviation of most private ´¬ürms is un-
observable since there are no reliable market prices. Therefore, we must